Certified and accurate computation of function space norms of deep neural networks

TL;DR

This paper develops a framework for certified, reliable computation of function space norms of neural networks, crucial for PDEs, by combining interval arithmetic, adaptive refinement, and quadrature methods.

Contribution

It introduces a novel approach that exploits neural network structure to compute guaranteed bounds on function space norms, surpassing point evaluation limitations.

Findings

Provides guaranteed bounds for Lebesgue and Sobolev norms of neural networks.

Establishes a convergence theorem for certified adaptive quadrature procedures.

Demonstrates practical certified bounds for PINN residuals through numerical experiments.

Abstract

Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do not provide enough information to derive tight deterministic and guaranteed bounds on function space norms. In this work, we move beyond a purely black-box setting and exploit the neural network structure directly. We present a framework for the certified and accurate computation of integral quantities of neural networks, including Lebesgue and Sobolev norms, by combining interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement and quadrature-based aggregation. On each box, we compute guaranteed lower and upper bounds for function values and derivatives, and propagate these local certificates to global lower and upper bounds for the target integrals. Our analysis provides a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, yielding certified computation of $L^p$, $W^{1,p}$, and $W^{2,p}$ norms. We further show how these ingredients lead to practical certified bounds for PINN interior residuals. Numerical experiments illustrate the accuracy and practical behavior of the proposed methods.